Epsilon-expansion in the N-component ϕ 4 model

The formalism of projection Hamiltonians is applied to the N-component O(N)-invariant ϕ⁴ model in the Euclidean and p-adic spaces. We use two versions of the ε-expansion (with ε = 4 − d and with ε = α − 3d/2, where α is the renormalization group parameter) and evaluate the critical indices ν and η up to the second order of the perturbation theory. The results for the (4− d)-expansion then coincide with the known results obtained via the quantum-field renormalization-group methods. Our calculations give evidence that in dimension three, both expansions describe the same non-Gaussian fixed point of the renormalization group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 365–384, March, 2006.     

Concentration-cancellation phenomena for the velocity fields in incompressible fluid flows

A more intuitive sufficient condition is given for the concentration cancellation phenomena in 2- or 3-D incompressible fluid flows; that is, if the projection of concentration set of the weak-star defect measure associated with the approximate solution sequence onto space ℝ^π _x (n = 2, 3) is a set with Hausdorff dimension less than 1, then the weak-L ² limit of the approximate solution sequence is a classical weak solution of Euler equation. Using this condition, an example is given to elucidate concentration-cancellation phenomena.     

On the solutions of the translation equation in rings of formal power series

We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing homomorphisms Ф from (G, +) into the differential group Г¹∞ describing the chain rules of higher order of C∞ functions with fixed point 0. In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L¹∞),Ф = (f_n ⁿ≤1,forwhich f₁ = l, f₂ = ... = f_p+l =0,f_p+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w _n ^p )_n≥p+2 of universal polynomials in f_p+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w _n ^p )_n≥p+2 we may also use another sequence (L _n ^p )_n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions.     

Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).     

On the space l P(β)

Let be a sequence of positive numbers and 1 ≤p < ∞. We consider the spacel ^P(β) of all power series such that . We give a necessary and sufficient condition for a polynomial to be cyclic inl ^P(β) and a point to be bounded point evaluation onl ^P(β).     

Spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator

Estimate for the remainder in the Weyl asymptotics of the spectrum of the Maxwell operator in Lipschitz domains

We study the asymptotics of the spectrum of the Maxwell operator M in a bounded Lipschitz domain W ì \mathbbR³ \Omega \subset {\mathbb{R}^3} under the condition of the perfect conductivity of the boundary ∂Ω. We obtain the following estimate for the remainder in the Weyl asymptotic expansion of the counting function N(λ,M) of positive eigenvalues of the Maxwell operator M:

A joint norm control Nehari type theorem forN-tuples of Hardy spaces

If N ∈ ℕ, 0 < p ≤ 1, and(X_k) _k=1 ^N are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩ _k=1 ^N X_k, there exists with , for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ², it is proved that the N-tuple (H(X₁),…, H(X_n)) is K_⊓-closed with respect to (X₁,…, X_N) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result.     

When is a Markovian Semigroup Induced by a Semiflow?

t , for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating the points of X and satisfying Zf² = 2fZf. If either (1) there exists δ > 0 such that (T^t f)²∈ D if 0 ≤ t ≤δ for each f in this family; or (1′) for some core D′ of Z, g ∈ D′ implies g²∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition) of continuous functions φ_t from X into X — such that T^tf(x) = f(φ_t (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating family satisfying Zf² = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions are redundant. If the functions φ_t are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined by < f, g > = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup and is equivalently given by N = {f ∈ D| f²∈ D and Zf² = 2fZf} = {f ∈ D| T^t(f²)-(T^tf)² is o(t²) in C(X)}. In the general case N is an algebra closed under composition with any C¹-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as a derivation) is a special case of a theorem for C²-functions φ which holds more generally for all f in d, viz., Z(φ[f] = (Zf) φ′[f] + ? <f, f> φ″[f], Provided Z is a local operator and D is an algebra. In this case the form < f, g > itself enjoys the relation < φ[f], ψ[g] > = φ′ [f] ψ′[g] < f, g >, for C²functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization of Markovian semigroups that are groups: Let T^t = e^tz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||T^t - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a²) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the self-adjoint elements of A. The e^tz , -∞ < t < ∞, is a group of Markovian operators.     

The self-similar solutions of the Tricomi-type equations

This paper analyses the properties of the family of self-similar solutions of the generalized Tricomi equation u_tt - t^2k Du = 0 (2k ? \mathbbN)u_{{tt}} - t^{{2k}} \Delta u = 0\,(2k \in {{\mathbb{N}}}) in the domain \mathbbR ₊ ^{1 + n}{{\mathbb{R}}}_{ + }^{{1 + n}} by considering initial conditions on the functions and their derivatives, posed as the Cauchy problem with homogeneous initial data. For specific values of the power k ( = 1/2 or = 3/2) and n = 1 this problem has applications in the aerodynamics of airfoils operating in transonic flows of perfect or dense gases, respectively. An integral transformation is suggested and used to represent the solutions of the Cauchy problem with homogeneous initial functions in terms of fundamental solutions of the classical wave equation (the case k = 0). Then the Cauchy problem with homogeneous initial functions for the wave equation in \mathbbR^{1 + n}{{\mathbb{R}}}^{{1 + n}} is solved. These results are used to derive estimates of the upper bound for solutions’ size and to obtain the asymptotics for self-similar solutions of the wave equation and of the Tricomi-type equation in the neighbourhood of their light cones.     

Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

Lucas's criterion for the primality of numbers of the form N = h2n−1

The following theorem is proved. Let N = h2ⁿ-1, where n ≥ 2, h is odd, 1 <-h < 2ⁿ, and suppose that v is a positive integer, v ≥ 3,α is a root of the equation $$(v^2 - 4,N) = 1,\left( {\frac{{v - 2}}{N}} \right) = 1,\left( {\frac{{v + 2}}{N}} \right) = - 1$$ . Then for N to be prime, it is necessary and sufficient that s_n?2≡0(modN), where S_k+1=S _k ² ? 2 (k = 0, 1...), s_o=a^h+ a^?h. For given N, an algorithm is described for the construction of the smallest v satisfying the conditions of this theorem.     

Local convexity of the growth function of finitely generated groups and question 5.2 in the Kourovka Notebook

Using the group <a,b|a³=b³=(ab)³=1>, we refute the conjecture dubbed in 1976 by V. Belyaev and N. Sesekin, which maintained that the growth function σ(n) of a finitely generated group satisfies the inequality σ(n)≤(σ(n−1)+σ(n+1))/2 for all sufficiently large n. Supported by the National Research Foundation of Switzerland, and by RFFR grant No. 96-01-00974. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 621–626, November–December, 1998.     

The initial boundary-value problem with a free surface condition for the penalized equations of aqueous solutions of polymers

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (ℝ⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω⊂ℝ³. Bibliography: 25 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 185–207. Translated by N. A. Karazeeva.     

Estimates of the number of singular points of a complex hypersurface and related questions

It is well known that the number of isolated singular points of a hypersurface of degree d in ℂP^m does not exceed the Arnol’d number A_m(d), which is defined in combinatorial terms. In the paper it is proved that if b _m−1 ^± (d) are the inertia indices of the intersection form of a nonsingular hypersurface of degree d in ℂP^m, then the inequality A_m(d)<min{b _m−1 ⁺ (d), b _m−1 ⁻ (d)} holds if and only if (m−5)(d−2)≥18 and (m,d)≠(7,12). The table of the Arnol’d numbers for 3≤m≤14, 3≤d≤17 and for 3≤m≤14, d=18, 19 is given. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 180–190. Translated by O. A. Ivanov and N. Yu. Netsvetev.     

On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \mathbb{R}^{N}

In this paper we establish a Serrin’s type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equations in It is proved that if the gradient of pressure belongs to L^{α, γ} with then the weak solution actually is regular and unique. Received: May 4, 2004     

On the general additive divisor problem

We obtain a new upper bound for the sum Σ _h≤H Δ _k (N, h) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ _k (N, h) is the (expected) error term in the asymptotic formula for Σ _N<n≤2N d _k (n)d _k (n + h), and d _k (n) is the divisor function generated by ζ(s) ^k . When k = 3, the result improves, for H ≥ N ^1/2, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3.     

A note on the Lehmer problem over short intervals

Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p ^1/2log² p). The main purpose of this paper is to give an asymptotic formula for Σ _c=1 ^p−1(r(N, c; p) − 1/2N)².     

On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.